Adhesion toughness of multilayer graphene films

Interface adhesion toughness between multilayer graphene films and substrates is a major concern for their integration into functional devices. Results from the circular blister test, however, display seemingly anomalous behaviour as adhesion toughness depends on number of graphene layers. Here we show that interlayer shearing and sliding near the blister crack tip, caused by the transition from membrane stretching to combined bending, stretching and through-thickness shearing, decreases fracture mode mixity G II/G I, leading to lower adhesion toughness. For silicon oxide substrate and pressure loading, mode mixity decreases from 232% for monolayer films to 130% for multilayer films, causing the adhesion toughness G c to decrease from 0.424 J m−2 to 0.365 J m−2. The mode I and II adhesion toughnesses are found to be G Ic = 0.230 J m−2 and G IIc = 0.666 J m−2, respectively. With point loading, mode mixity decreases from 741% for monolayer films to 262% for multilayer films, while the adhesion toughness G c decreases from 0.543 J m−2 to 0.438 J m−2.

2. In the supplementary text, the authors go from equation S1 to S20 by simply setting the width of the beam to a unit value. It is not quite clear how this can be done given that equation S1 is for a beam while equation S20 is supposed to be for a circular plate. It would be helpful if the authors can elaborate on this.
3. The author's need to elaborate on sliding of layers in multilayer graphene. When they say sliding, it is not quite clear if it is restricted to the delaminated region or extends to adhered region too. Also, can they comment for each case how the analysis changes, if it does? The authors define the fracture mode mixity as G_I/G_II, which differs from the common definition used in fracture mechanics literature. More commonly, we would define the mode mixity as G_II/G_I, so that the mode I has the lowest mode mixity and typically the lowest toughness as well. It appears unnecessary to make a difference here.
There seems to a typo in the reference number for He et al in Page 2. Should be 5 instead of 4, according to the reference list at the end. However, Zong et al., listed as reference no. 4, is not mentioned until much later in the text. Perhaps the reference list should be re-ordered.
In Page 3, the authors claim that "fracture mode mixity and the sliding effect are not considered anywhere in the current mechanical models". However, as noted above, the effect has been considered by Cao et al (although for different experiments) and discussed in the review paper by Akinwande et al.
Equation (1) looks similar to that of Hencky's solution as used by Koenig  A major question is concerning the partition of the energy release rate into mode I and mode II as given in Eqs. 7 and 8, with more details in the SI. The method is based on the authors' previous works (Refs. 14- 18). First of all, this method has not been well accepted by others in the field, and there appears to be serious dispute over its validity. Second, with all information provided by the authors (including SI), it is very difficult to judge the soundness of this partition for anyone who is not specialized in this particular problem (including this reviewer). One would have to go to the authors' previous papers to dig out the essence behind the equations. Even so (as this reviewer did), it is not an easy task. To the best of this reviewer's effort, this partition seems questionable if not flawed. To convince the readers, the authors would have to deliver the message more effectively. For example, how do they define G_I and G_II (before they reached Eqs. S1 and S2)? It is well known that the stress field for an interfacial crack in general cannot be separated into mode I and mode II (unless a cohesive zone model is used).
Another question is concerning the parameter \lambda in Eq. 7, which is said to represent the effect of sliding. It is unclear how this effect is taken into account. Eq. (S39) defines the parameter following some lengthy mathematics (which is very difficult to follow through), but it does not help understanding how the effect of sliding is treated. Is there any relative sliding displacement between the layers (ahead of the crack tip)? Is there a critical shear stress for sliding to occur?
Moreover, a correction factor S(n) is introduced in Eq. 10 and given in Eq. 13. This appears to be completely ad hoc, in order to match the experimental data. This again casts some doubt on the partition method.
Is Eq. 22 true for all cases? It seems to depend on the parameter \lambda, which in turn depends on the number of layers.

Response to reviewers:
Reviewer 1

1.1.
The authors need to correlate the physical mechanism of sliding to the parameter lambda in the manuscript, which was introduced as a dimensionless parameter indicating the level of sliding.
• In the main article on p. 5: The λ parameter in equations (7) and (9) represents the interlayer shearing and sliding effect at the blister crack tip, which is given as • In the main article on p. 6: In the case of monolayer graphene films, the shear force in equation (5) makes no contribution to the ERR in the membrane limit because there is no interlayer shearing and sliding. In the case of multilayer graphene films, interlayer shearing and sliding occurs near the blister crack tip, caused by the transition from membrane stretching to combined bending, stretching and through-thickness shearing. Consequently, interlayer shearing and sliding activates the shear force in equation (5). Its action is introduced through the λ parameter in conjunction with the interlayer shearing and sliding factor ( ) n S , which is assumed to take the following form: • In the supplementary information on pp. 9-11: The origin of the parameter λ is obvious, but the origin of the factor ( ) shear force has no effect on ERR in the membrane limit for monolayer films.
Multilayer graphene membranes 18 are considered next. As before, for linear bending at small deflection, through-thickness shear force exists and produces through-thickness strain, resulting in extra ERR. In the membrane limit, it is expected that multilayer graphene films in the membrane region of a blister behave as a single layer because there is only membrane stretching. The transition from membrane stretching to combined bending, stretching and through-thickness shearing occurs near the crack tip. If a multilayer graphene film still behaves as a single layer in the transition region, as is the case for monolayer graphene membranes, the through-thickness shearing strain energy near the crack tip is still negligible, resulting in no ERR contribution; however, the transition can cause interlayer shearing and sliding in the transition region, at the crack tip in particular, the through-thickness shearing must account for the fact mentioned above that adhesion toughness has a large decrease between monolayer and two-layer graphene films and remains fairly constant afterwards 18 .
(3) From the view point of continuum mechanics, the converged value of ( ) n S for multilayer graphene films is assumed to make a complete transition from membrane stretching to combined bending, stretching and shearing at the crack tip, so ( ) 1

= ∞ S
. The validity of ( ) Tables S1-S15. There is a large decrease from monolayer to two-layer graphene films and then only a small decrease from two-layer to three-layer graphene films. For the three-, fourand five-layer graphene films, the values of the λ parameter are very close to each other.
This shows the typical interlayer shearing and sliding behaviour.

The authors need to confirm the values of lambda used in the manuscript can predict reasonable value for interfacial shear stress between graphene layers.
• This comment is related to Comment 1.1 as two perspectives on the same issue: Comment 1.1 relates and the interlayer shearing and sliding while this comment relates to and the interlayer shear stress. Therefore, please refer to our response to Comment 1.1.
• In the supplementary information on p. 11: In addition to above explanation of ( ) is to introduce an effective through-thickness shear modulus S G to account for the interlayer shearing and sliding. Note that S G is just an effective value instead of the actual material property. This is similar to case of classical plate theory in which the effective through-thickness shear modulus is infinitely large while the actual material property is finite.
• As noted in the bullet point above, the parameter represents the general averaged effect of interlayer shearing and sliding, but the interlayer shear stress is a local quantity, and in particular, the shear stress varies from interlayer to interlayer. Consequently, the present approach is unable to calculate the shear stress.
• In the supplementary information on p. 11: Finally, the critical interlayer shear stress for sliding is beyond the scope of the present work; however, the present methodology can be used to determine the mode I and II toughness between graphene layers using the blister test.
The mode II toughness is considered to be the sliding toughness.

As noted in Fig. 2a in the manuscript, the value of lambda is different from two layers to three layers of graphene membrane. How to understand this difference if assuming the shear between graphene layers is a constant?
The changing value of is understood through the physical mechanism of shearing and sliding (which is the subject of Comment 1.1., above), which is has now been more thoroughly explained in the supplementary information on pp. 9-11. See the response given to Comment 1.1., above.    • Throughout the main article and the supplementary information, the sliding is now referred to as "interlayer shearing and sliding near the blister crack tip".
• In the supplementary information on p. 11: Note that the interface between graphene films and their substrates is assumed to be a rigid interface 1-5,7-12 , that is, it is assumed that no relative shearing and sliding displacement occurs before separation. This is consistent with Koenig et al.'s 18 work. The present methodology could, however, be extended to consider the shearing and sliding analytically by combining it with the authors' mixed-mode partition theory for non-rigid interface fractures 6 . Some complex mechanical behaviour such as wrinkling 30 can be caused by this type sliding, which will be considered in future work. Fig. 2.

Please check the units in the axis labels for and in
The units of and in Fig. 2e and f and Fig. 3 have been corrected to µm.   The reviewer is correct: References 4 and 5 were listed in the reverse order. This has been fixed now.

Equation (1) looks similar to that of Hencky's solution as used by Koenig et al and that of a simpler solution by Yue et al (Ref. 8), but the dependence on
Poisson's ratio in Eq. (2) is different. Please briefly comment on the reason for this difference.

A major question is concerning the partition of the energy release rate into mode I and mode II as given in Eqs. 7 and 8, with more details in the SI. The method is based on the authors' previous works (Refs. 14-18). First of all, this method has not been well accepted by others in the field, and there appears to be serious dispute over its validity. Second, with all information provided by the authors (including SI), it is very difficult to judge the soundness of this partition
for anyone who is not specialized in this particular problem (including this reviewer). One would have to go to the authors' previous papers to dig out the essence behind the equations. Even so (as this reviewer did), it is not an easy task. To the best of this reviewer's effort, this partition seems questionable if not flawed. To convince the readers, the authors would have to deliver the message more effectively. For example, how do they define G_I and G_II (before they reached Eqs. S1 and S2)? It is well known that the stress field for an interfacial crack in general cannot be separated into mode I and mode II (unless a cohesive zone model is used).  Equations (S1) and (S2) can be written in the following forms 7-10 based on 2D elasticity: Furthermore, in the case of isotropic materials, equations (S5) and (S6) reduce to The pure modes, 2D (S8) are reduced to the case of thin films in the blister test to determine the adhesion toughness, for example, the adhesion toughness of multilayer graphene films 18 . The substrate is treated as infinitely thick and the films as very thin, as shown in Fig. S2a; therefore, the thickness ratio tends to infinity ∞ → γ . The authors' latest work on the mechanical behaviour of thin film spallation [14][15][16][17] shows that excellent agreement is achieved with experimental results 19-23 when the material mismatch between a film and its substrate is neglected. Furthermore, in these studies 14,15 slightly worse agreement was found with experimental results 19,20 when the mismatch 8,9 was taken into account. Therefore, the present work also neglects the material mismatch, and equations (S7) and (S8) become where 1 h is the film thickness. Note that the authors' pure modes as it is believed to be more accurate.

Another question is concerning the parameter \lambda in Eq. 7, which is said to represent the effect of sliding. It is unclear how this effect is taken into account.
Eq. (S39) defines the parameter following some lengthy mathematics (which is very difficult to follow through), but it does not help understanding how the effect of sliding is treated. Is there any relative sliding displacement between the layers (ahead of the crack tip)? Is there a critical shear stress for sliding to occur?
• Reviewer 1 asks similar questions regarding . A detailed response is given in our responses to comments 1.1, 1.2 and 1.3 (above).
• In the supplementary information on p. 11: Note that the interface between graphene films and their substrates is assumed to be a rigid interface 1-5,7-12 , that is, it is assumed that no relative shearing and sliding displacement occurs before separation. This is consistent with Koenig et al.'s 18 work. The present methodology could, however, be extended to consider the shearing and sliding analytically by combining it with the authors' mixed-mode partition theory for non-rigid interface fractures 6 . Some complex mechanical behaviour such as wrinkling 30 can be caused by this type sliding, which will be considered in future work.
• In the supplementary information on p. 11: Finally, the critical interlayer shear stress for sliding is beyond the scope of the present work; however, the present methodology can be used to determine the mode I and II toughness between graphene layers using the blister test.
The mode II toughness is considered to be the sliding toughness.

Moreover, a correction factor S(n) is introduced in Eq. 10 and given in Eq.
13. This appears to be completely ad hoc, in order to match the experimental data. This again casts some doubt on the partition method.
Reviewer 1 asks similar questions regarding (and consequently ( )). A detailed response is given in our response to comments 1.1, 1.2 and 1.3 (above).

Reviewers' Comments:
Reviewer #1: Remarks to the Author: The current reviewer noticed that Reviewer #3 raised two important questions, and would like to add more comments.
(1) Reviewer #3 questioned the validity on the partition of energy release rate into mode I and mode II.
--This theory is based on the authors' previous works, which are not familiar to the majority of scholars in mechanics filed (including the current viewer and reviewer #3). Without significant effects, it is hard to justify the theory itself.
(2) Reviewer #3 suggested to deliver partition theory more effectively so that the audiences can follow the paper without dig into the authors' previous publications.
--The authors have not fully address this comment. The authors did not give sufficient background to the theory. Instead, they simply claimed the theory itself is well established and verified, which may not be true. More importantly, it should be noticed that the audiences will not learn a lot from this work, especially those equations in the SI, without the necessary background. One important feature of Nature Comm is to attract broader audiences, and thus to make significant impact in related fields. However, the current manuscript may not sever this purpose well enough. The manuscript may be more suitable for specialized Journals, where the main audiences are more interested in partition theory. Essentially, this work is an application of such a theory.
Reviewer #2: Remarks to the Author: All this reviewer's concerns have been adequately addressed. in which the validity is tested.

Response to reviewers
• Furthermore, regarding the validity of the partition theory, the theory has been presented in several high quality international conferences as invited plenary lectures. The latest plenary lecture was presented in June 2017 at the 14th International Conference on Fracture (ICF14) which is regarded as the top conference on fracture mechanics, held every 4 years, and most world-leading fracture mechanics experts attended. In addition, several technical sessions were invited to be organised to report the work, including in ICF14, and in August 2017 at the 21st International Conference on Composite Materials (ICCM21).
• Additions (highlighted in yellow) have been made to the Supplementary Information as follows: • In the Supplementary Information on p. 3 Equations (S1) and (S2) can be written in the following forms 7-10 based on 2D elasticity: • Furthermore, the aim of this work was to develop a mechanical model and methodology to correctly determine the adhesion toughness between multilayer graphene films and substrates. Even if the work was just an application of the partition theory (which it isn't), the aim is nevertheless achieved, and this even on its own, we believe, would make it a valuable contribution to the field. The thoroughly-derived analytical formulae and methodology can, for the first time, be readily used and accessed by broader audiences. We therefore believe it will make a significant impact in the field.
• In the Supplementary Information on p. 5: In the following, equations (S7) and (S8) are extended to the case of thin films in the blister test to determine the adhesion toughness, for example, the adhesion toughness of multilayer graphene films 19 .